Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

2019 ◽  
Vol 19 (02) ◽  
pp. 1950010 ◽  
Author(s):  
Libor Barto ◽  
Michael Kompatscher ◽  
Miroslav Olšák ◽  
Van Pham Trung ◽  
Michael Pinsker
Keyword(s):  
Constraint Satisfaction ◽  
Short Proof ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Ramsey Property ◽  
Categorical Structure ◽  
Homogeneous Structures ◽  
Identity Modulo ◽  
Rational Order ◽  
Its Polymorphism

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain nontrivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being nontrivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that [Formula: see text]-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking another approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a nontrivial system of linear identities, and obtain nontrivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every [Formula: see text]-categorical structure is homomorphically equivalent to a model-complete core.

scholarly journals Constraint Satisfaction Problems and Their Reduction to Directed Graphs

2021 ◽  
Author(s):  
Muhanda Stella Mbaka Muzalal
Keyword(s):  
Constraint Satisfaction ◽  
Cell Phone ◽  
Constraint Satisfaction Problem ◽  
Directed Graphs ◽  
Oriented Graph ◽  
Constraint Satisfaction Problems ◽  
Systems Of Equations ◽  
Relational Structures ◽  
Algorithmic Problems ◽  
Boolean Formulas

Constraint satisfaction problems present a general framework for studying a large class of algorithmic problems such as satisfaction of Boolean formulas, solving systems of equations over finite fields, graph colourings, as well as various applied problems in artificial intelligence (scheduling, allocation of cell phone frequencies, among others.) CSP (Constraint Satisfaction Problems) bring together graph theory, complexity theory and universal algebra. It is a well known result, due to Feder and Vardi, that any constraint satisfaction problem over a finite relational structure can be reduced to the homomorphism problem for a finite oriented graph. Until recently, it was not known whether this reduction preserves the type of the algorithm which solves the original constraint satisfaction problem, so that the same algorithm solves the corresponding digraph homomorphism problem. We look at how a recent construction due to Bulin, Deli´c, Jackson, and Niven can be used to show that the polynomial solvability of a constraint satisfaction problem using Datalog, a programming language which is a weaker version of Prolog, translates from arbitrary relational structures to digraphs.

Symbolic Solutions for Symbolic Constraint Satisfaction Problems

2020 ◽  
Author(s):  
Alexsander Andrade de Melo ◽  
Mateus De Oliveira Oliveira
Keyword(s):  
Constraint Satisfaction ◽  
Existence Of Solutions ◽  
Constraint Satisfaction Problem ◽  
Constant Width ◽  
Computable Function ◽  
Constraint Satisfaction Problems ◽  
Decision Diagrams ◽  
Turing Machines ◽  
Branching Programs ◽  
Hard Problems

A fundamental drawback that arises when one is faced with the task of deterministically certifying solutions to computational problems in PSPACE is the fact that witnesses may have superpolynomial size, assuming that NP is not equal to PSPACE. Therefore, the complexity of such a deterministic verifier may already be super-polynomially lower-bounded by the size of a witness. In this work, we introduce a new symbolic framework to address this drawback. More precisely, we introduce a PSPACE-hard notion of symbolic constraint satisfaction problem where both instances and solutions for these instances are implicitly represented by ordered decision diagrams (i.e. read-once, oblivious, branching programs). Our main result states that given an ordered decision diagram D of length k and width w specifying a CSP instance, one can determine in time f(w,w')*k whether there is an ODD of width at most w' encoding a solution for this instance. Intuitively, while the parameter w quantifies the complexity of the instance, the parameter w' quantifies the complexity of a prospective solution. We show that CSPs of constant width can be used to formalize natural PSPACE hard problems, such as reachability of configurations for Turing machines working in nondeterministic linear space. For such problems, our main result immediately yields an algorithm that determines the existence of solutions of width w in time g(w)*n, where g:N->N is a suitable computable function, and n is the size of the input.

PREPROCESSING QUANTIFIED CONSTRAINT SATISFACTION PROBLEMS WITH VALUE REORDERING AND DIRECTIONAL ARC AND PATH CONSISTENCY

2008 ◽  
Vol 17 (02) ◽  
pp. 321-337 ◽  
Author(s):  
KOSTAS STERGIOU
Keyword(s):  
Constraint Satisfaction ◽  
Early Stage ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Combinatorial Problems ◽  
Arc Consistency ◽  
Solution Methods ◽  
Speed Up ◽  
Phase Transition Region ◽  
The Impact

The Quantified Constraint Satisfaction Problem (QCSP) is an extension of the CSP that can be used to model combinatorial problems containing contingency or uncertainty. It allows for universally quantified variables that can model uncertain actions and events, such as the unknown weather for a future party, or an opponent's next move in a game. Although interest in QCSPs is increasing in recent years, the development of techniques for handling QCSPs is still at an early stage. For example, although it is well known that local consistencies are of primary importance in CSPs, only arc consistency has been extended to quantified problems. In this paper we contribute towards the development of solution methods for QCSPs in two ways. First, by extending directional arc and path consistency, two popular local consistencies in constraint satisfaction, to the quantified case and proposing an algorithm that achieves these consistencies. Second, by showing how value ordering heuristics can be utilized to speed up computation in QCSPs. We study the impact of preprocessing QCSPs with value reordering and directional quantified arc and path consistency by running experiments on randomly generated problems. Results show that our preprocessing methods can significantly speed up the QCSP solving process, especially on hard instances from the phase transition region.

scholarly journals An Ant Colony Optimization Based on Information Entropy for Constraint Satisfaction Problems

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 766 ◽  
Author(s):  
Boxin Guan ◽  
Yuhai Zhao ◽  
Yuan Li
Keyword(s):  
Local Search ◽  
Ant Colony Optimization ◽  
Constraint Satisfaction ◽  
Information Entropy ◽  
Hypothesis Test ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Ant Colony ◽  
Cost Comparison ◽  
Solution Quality

Solving the constraint satisfaction problem (CSP) is to find an assignment of values to variables that satisfies a set of constraints. Ant colony optimization (ACO) is an efficient algorithm for solving CSPs. However, the existing ACO-based algorithms suffer from the constructed assignment with high cost. To improve the solution quality of ACO for solving CSPs, an ant colony optimization based on information entropy (ACOE) is proposed in this paper. The proposed algorithm can automatically call a crossover-based local search according to real-time information entropy. We first describe ACOE for solving CSPs and show how it constructs assignments. Then, we use a ranking-based strategy to update the pheromone, which weights the pheromone according to the rank of these ants. Furthermore, we introduce the crossover-based local search that uses a crossover operation to optimize the current best assignment. Finally, we compare ACOE with seven algorithms on binary CSPs. The experimental results revealed that our method outperformed the other compared algorithms in terms of the cost comparison, data distribution, convergence performance, and hypothesis test.

scholarly journals Solving Weighted Constraint Satisfaction Problems with Memetic/Exact Hybrid Algorithms

2009 ◽  
Vol 35 ◽  
pp. 533-555 ◽  
Author(s):  
J. E. Gallardo ◽  
C. Cotta ◽  
A. J. Fernández
Keyword(s):  
Constraint Satisfaction ◽  
Large Scale ◽  
Constraint Satisfaction Problem ◽  
Heuristic Method ◽  
Optimal Solution ◽  
Constraint Satisfaction Problems ◽  
Still Life ◽  
Level Model ◽  
Weighted Constraint

A weighted constraint satisfaction problem (WCSP) is a constraint satisfaction problem in which preferences among solutions can be expressed. Bucket elimination is a complete technique commonly used to solve this kind of constraint satisfaction problem. When the memory required to apply bucket elimination is too high, a heuristic method based on it (denominated mini-buckets) can be used to calculate bounds for the optimal solution. Nevertheless, the curse of dimensionality makes these techniques impractical on large scale problems. In response to this situation, we present a memetic algorithm for WCSPs in which bucket elimination is used as a mechanism for recombining solutions, providing the best possible child from the parental set. Subsequently, a multi-level model in which this exact/metaheuristic hybrid is further hybridized with branch-and-bound techniques and mini-buckets is studied. As a case study, we have applied these algorithms to the resolution of the maximum density still life problem, a hard constraint optimization problem based on Conway's game of life. The resulting algorithm consistently finds optimal patterns for up to date solved instances in less time than current approaches. Moreover, it is shown that this proposal provides new best known solutions for very large instances.

scholarly journals On the Subexponential-Time Complexity of CSP

2015 ◽  
Vol 52 ◽  
pp. 203-234 ◽  
Author(s):  
Ronald De Haan ◽  
Iyad Kanj ◽  
Stefan Szeider
Keyword(s):  
Constraint Satisfaction ◽  
Time Complexity ◽  
Constraint Satisfaction Problem ◽  
Structural Parameters ◽  
Constraint Satisfaction Problems ◽  
Global Constraints ◽  
Brute Force ◽  
Subexponential Time ◽  
Complete Problems ◽  
Np Complete

Not all NP-complete problems share the same practical hardness with respect to exact computation. Whereas some NP-complete problems are amenable to efficient computational methods, others are yet to show any such sign. It becomes a major challenge to develop a theoretical framework that is more fine-grained than the theory of NP-completeness, and that can explain the distinction between the exact complexities of various NP-complete problems. This distinction is highly relevant for constraint satisfaction problems under natural restrictions, where various shades of hardness can be observed in practice. Acknowledging the NP-hardness of such problems, one has to look beyond polynomial time computation. The theory of subexponential-time complexity provides such a framework, and has been enjoying increasing popularity in complexity theory. An instance of the constraint satisfaction problem with n variables over a domain of d values can be solved by brute-force in dn steps (omitting a polynomial factor). In this paper we study the existence of subexponential-time algorithms, that is, algorithms running in do(n) steps, for various natural restrictions of the constraint satisfaction problem. We consider both the constraint satisfaction problem in which all the constraints are given extensionally as tables, and that in which all the constraints are given intensionally in the form of global constraints. We provide tight characterizations of the subexponential-time complexity of the aforementioned problems with respect to several natural structural parameters, which allows us to draw a detailed landscape of the subexponential-time complexity of the constraint satisfaction problem. Our analysis provides fundamental results indicating whether and when one can significantly improve on the brute-force search approach for solving the constraint satisfaction problem.

scholarly journals Nature-Inspired Techniques For Dynamic Constraint Satisfaction Problems

2021 ◽  
Author(s):  
Mehdi Bidar ◽  
Malek Mouhoub
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Dynamic Environment ◽  
Iterative Algorithms ◽  
Constraint Satisfaction Problems ◽  
Environmental Stability ◽  
Final State ◽  
Dynamic Constraint ◽  
Consistent Solution ◽  
Constraint Problem

Abstract Combinatorial applications such as configuration, transportation and resource allocation, often operate under highly dynamic and unpredictable environments. In this regard, one of the main challenges is to maintain a consistent solution anytime constraints are (dynamically) added. While many solvers have been developed to tackle these applications, they often work under idealized assumptions of environmental stability. In order to address limitation, we propose a methodology, relying on nature-inspired techniques, for solving constraint problems when constraints are added dynamically. The choice for nature-inspired techniques is motivated by the fact that these are iterative algorithms, capable of maintaining a set of promising solutions, at each iteration. Our methodology takes advantage of these two properties, as follows. We first solve the initial constraint problem and save the final state (and the related population) after obtaining a consistent solution. This saved context will then be used as a resume point for finding, in an incremental manner, new solutions to subsequent variants of the problem, anytime new constraints are added. More precisely, once a solution is found, we resume from the current state to search for a new one (if the old solution is no longer feasible), when new constraints are added. This can be seen as an optimization problem where we look for a new feasible solution satisfying old and new constraints, while minimizing the differences with the solution of the previous problem, in sequence. This latter objective ensures to find the least disruptive solution, as this is very important in many applications including scheduling, planning and timetabling. Following on our proposed methodology, we have developed the dynamic variant of several nature-inspired techniques to tackle dynamic constraint problems. Constraint problems are represented using the well-known Constraint Satisfaction Problem (CSP) paradigm. Dealing with constraint additions in a dynamic environment can then be expressed as a series of static CSPs, each resulting from a change in the previous one by adding new constraints. This sequence of CSPs is called the Dynamic CSP (DCSP). To assess the performance of our proposed methodology, we conducted several experiments on randomly generated DCSP instances, following the RB model. The results of the experiments are reported and discussed.

A SPACE-EFFICIENT BACKTRACK-FREE REPRESENTATION FOR CONSTRAINT SATISFACTION PROBLEMS

2008 ◽  
Vol 17 (04) ◽  
pp. 703-730 ◽  
Author(s):  
J. CHRISTOPHER BECK ◽  
TOM CARCHRAE ◽  
EUGENE C. FREUDER ◽  
GEORG RINGWELSKI
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Significant Proportion ◽  
Constraint Satisfaction Problems ◽  
Basic Algorithm ◽  
High Coverage ◽  
Trade Off ◽  
Free Representation ◽  
Pruning Heuristics ◽  
First Time

In this paper we present a radical approach to obtaining a backtrack-free representation for a constraint satisfaction problem: remove values that lead to dead-ends. This technique does not require additional space but has the drawback of removing solutions. We investigate a number of variations on the basic algorithm including the use of seed solutions, consistency techniques, and a variety of pruning heuristics. Our experimental results indicate that a significant proportion of the solutions to the original problem can be retained especially when an optimization algorithm that specifically searches for such “good” backtrack-free representations is employed. Further extensions increase solution retention by searching for high-coverage backtrack-free representations, by removing tuples rather than values, and by combining multiple backtrack-free representations. Our approach elucidates, for the first time, a three-way trade-off between space complexity, potential backtracks, and solution loss and enables algorithms that can actively reason about the trade-off between space, backtracks, and solution loss.

Constraint Processing

2011 ◽  
pp. 404-409
Author(s):  
Roman Barták
Keyword(s):  
Logic Programming ◽  
Constraint Satisfaction ◽  
Optimization Problems ◽  
Constraint Satisfaction Problem ◽  
Constraint Logic Programming ◽  
Constraint Satisfaction Problems ◽  
Start Time ◽  
Constraint Processing ◽  
Available Technology ◽  
3D Scene

Constraints appear in many areas of human endeavour starting from puzzles like crosswords (the words can only overlap at the same letter) and recently popular Sudoku (no number appears twice in a row) through everyday problems such as planning a meeting (the meeting room must accommodate all participants) till solving hard optimization problems for example in manufacturing scheduling (a job must finish before another job). Though all these problems look like being from completely different worlds, they all share a similar base – the task is to find values of decision variables, such as the start time of the job or the position of the number at a board, respecting given constraints. This problem is called a Constraint Satisfaction Problem (CSP). Constraint processing emerged from AI research in 1970s (Montanary, 1974) when problems such as scene labelling were studied (Waltz, 1975). The goal of scene labelling was to recognize a type of line (and then a type of object) in the 2D picture of a 3D scene. The possible types were convex, concave, and occluding lines and the combination of types was restricted at junctions of lines to be physically feasible. This scene labelling problem is probably the first problem formalised as a CSP and some techniques developed for solving this problem, namely arc consistency, are still in the core of constraint processing. Systematic use of constraints in programming systems has started in 1980s when researchers identified a similarity between unification in logic programming and constraint satisfaction (Gallaire, 1985) (Jaffar & Lassez, 1987). Constraint Logic Programming was born. Today Constraint Programming is a separate subject independent of the underlying programming language, though constraint logic programming still plays a prominent role thanks to natural integration of constraints into a logic programming framework. This article presents mainstream techniques for solving constraint satisfaction problems. These techniques stay behind the existing constraint solvers and their understanding is important to exploit fully the available technology.

Extending Fuzzy Constraint Satisfaction Problems

2006 ◽  
Vol 10 (4) ◽  
pp. 465-471 ◽  
Author(s):  
Yasuhiro Sudo ◽  
◽  
Masahito Kurihara ◽  
Tamotsu Mitamura ◽  
◽  
...  
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Computation Time ◽  
Approximate Solutions ◽  
Constraint Satisfaction Problems ◽  
Test Problems ◽  
Hybrid Domain ◽  
Fuzzy Constraint ◽  
Fuzzy Constraint Satisfaction ◽  
Discrete Domains

This paper propose a new type of Fuzzy CSP (Constraint Satisfaction Problem) that have a mixture of discrete and continuous domains, and a Spread-Repair algorithm. In traditional CSP and Fuzzy CSP, values for the variables are chosen from the discrete domains. However, this is often inconvenient when one wants to express real world problems. We show that this model, called HDFCSP (Hybrid Domain Fuzzy CSP), can be solved by Spread-Repair, an extension of the well known iterative improvement algorithms. Experimental results on some test problems show that the algorithm actually has an ability of finding partial approximate solutions with high probability in a computation time much shorter than the traditional, discrete-domain FCSP.

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